manifold learning

While contemporary self-supervised and masked/denoising autoencoder methods effectively learn strong representations from massive unlabeled data, their representational nature, cross-task generalization capability, and emergence mechanisms remain theoretically unexplained. Method: This project integrates statistical inference and nonconvex optimization theory to establish a unified analytical framework for unsupervised representation learning. Contribution/Results: It provides the first mathematical characterization of how self-supervised objectives—such as contrastive learning and reconstruction losses—induce structured latent spaces, and quantitatively links representation linear separability, invariance, and downstream generalization. The work identifies key theoretical conditions under which pretrained models achieve zero-shot transfer and task emergence in vision foundation models. Crucially, it delivers the first theoretical foundation for large-scale pretraining that is both statistically interpretable and optimization-traceable—bridging statistical guarantees with practical training dynamics.

Current unsupervised representation learning heavily relies on stationarity assumptions and topological priors, limiting its ability to model non-stationary, unstructured, high-dimensional signals prevalent in biological perception. To address this, we propose the first general-purpose unsupervised representation learning framework that requires neither stationarity nor topology priors. Our method integrates a learnable self-organizing layer, spectral clustering, and a masked autoencoder (MAE) to enable cross-modal adaptive representation learning. Evaluated on three highly heterogeneous, non-stationary data modalities—biologically inspired visual stimuli, neural electrophysiology recordings, and gene expression profiles—our approach significantly outperforms state-of-the-art methods including SimCLR and MAE. It establishes, for the first time, an unsupervised learning benchmark that combines theoretical universality with cross-modal robustness, offering a novel paradigm toward brain-like generalization capabilities.

High-dimensional scientific data often contain latent structures that are difficult to disentangle, and residual information—potentially encoding unmodeled physical signals—is frequently overlooked in generative modeling. Method: We propose a classifier-free-guided latent flow matching framework that explicitly decouples conditional information from residual representations, enabling interpretable extraction and controllable intervention of unmodeled scientific signals within generative models. By learning disentangled representations in latent space without label supervision, the method isolates semantically clear and physically interpretable features from the residual subspace. Results: Experiments on synthetic data, Colored MNIST, and the Galaxy10 astronomical dataset demonstrate substantial improvements in model analyzability and scientific interpretability. Our approach establishes a new paradigm for scientific discovery driven by high-dimensional data, bridging generative modeling with domain-informed, physics-aware analysis.

Existing hierarchical dimensionality reduction methods struggle to simultaneously preserve both local and global structures across multiple granularity levels while maintaining users’ mental map continuity. To address this, we propose HUMAP—the first hierarchical dimensionality reduction framework built upon UMAP. HUMAP achieves dual optimization of structural fidelity and mental map consistency through three core techniques: cross-scale neighborhood graph propagation, multi-granularity similarity modeling, and incremental embedding alignment. Compared to state-of-the-art methods, HUMAP significantly improves hierarchical structure preservation across multiple benchmark datasets. Furthermore, it demonstrates strong interpretability and interactive utility in real-world data labeling tasks. By unifying geometric structure preservation with cognitive consistency, HUMAP establishes a novel paradigm for multi-granularity visual analytics, enabling scalable, intuitive, and semantically grounded exploration of high-dimensional hierarchical data.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

Existing dimensionality reduction methods—such as t-SNE and UMAP—preserve local structure and yield well-separated clusters but often severely distort global geometry. To address this, we propose a modular visualization framework based on a “cluster-then-embed” paradigm: clustering (e.g., K-means, DBSCAN) and embedding (e.g., t-SNE, UMAP) are decoupled, and inter-cluster alignment constraints are explicitly imposed to govern the relative positions of local embeddings. This preserves local fidelity while restoring global structural consistency. The modular design enhances procedural transparency and result interpretability. Experiments on multiple synthetic and real-world datasets demonstrate that our method achieves visualization quality comparable to state-of-the-art approaches, while significantly improving global geometric fidelity and user interpretability.

Supervised deep learning suffers from “generalization collapse” on high-dimensional tabular data: while models perfectly fit the training distribution, they fail zero-shot out-of-distribution (OOD) anomaly detection. We identify the root cause as the absence of topological constraints in latent space, leading to manifold diffusion and indistinguishability between OOD samples and in-distribution data. To address this, we propose the first dual-stage explicit manifold learning framework. Stage I introduces a Dual-Centroid Compactness Loss to enforce compact, low-entropy hyperspherical clusters in latent space. Stage II builds a Masked Autoregressive Flow (MAF) density estimator atop this pre-structured manifold, decoupling manifold geometry learning from density modeling. Integrating a 1D-CNN with a Transformer encoder, our method achieves an F1-score of 0.87 on zero-shot evaluation of CIC-IDS-2017, with 88.89% detection rate for the Infiltration attack—substantially outperforming supervised baselines (0.00%) and the strongest unsupervised baseline (76.00%).

High-dimensional data—such as biological or image datasets—often exhibit substantial intra-class variance and nonlinear manifold structures, which challenge existing dimensionality reduction methods in effectively separating clusters and resolving subclusters. To address this, this work proposes MAPLE, a method that enhances UMAP’s manifold modeling capability through self-supervised learning and introduces Maximum-Margin Capacity Representations (MMCRs) to compress variance among locally similar points while amplifying dissimilarities between distinct ones. This yields a more accurate characterization of low-dimensional manifold geometry. Maintaining computational overhead comparable to UMAP, MAPLE significantly improves inter-cluster separation and subcluster resolution across diverse high-dimensional datasets, producing clearer and more refined visualizations.

High-dimensional data often reside on low-dimensional manifolds, yet existing manifold dimension estimation algorithms lack systematic evaluation and reproducible benchmarks. Method: We conduct a comprehensive empirical assessment of eight representative methods across synthetic and real-world datasets, quantifying the effects of noise, curvature, and sample size on estimation accuracy. We introduce a dataset-aware hyperparameter tuning principle and establish a controlled-variable experimental framework integrating local linear embedding, nearest-neighbor statistics, and multiscale geometric analysis. Contribution/Results: Contrary to the “complexity implies superiority” assumption, simple methods—such as the nearest-neighbor distance ratio and PCA-based gradient estimation—consistently achieve higher accuracy and robustness across most scenarios. This work delivers the first open-source, fully reproducible benchmark for manifold dimension estimation, accompanied by practical guidelines. It provides both theoretical insight and empirical evidence to inform unsupervised learning and dimensionality reduction methodology selection.

This work investigates the semantic evolution mechanisms and generalization origins of internal representations in deep neural networks (DNNs). We propose Gride—a nonparametric intrinsic dimension estimation framework based on nearest-neighbor distances—that robustly quantifies both the intrinsic dimensionality and uncertainty of hidden-layer representations. Integrating density-peak clustering with unsupervised topological analysis, we demonstrate that semantic hierarchies spontaneously emerge via density-based stratification. Our findings reveal that wide networks, even at zero training error, achieve regularization-driven generalization by leveraging redundant neurons to form low-dimensional, dense representation clusters. Furthermore, semantic structures across layers are fully recoverable in an unsupervised manner, and the topological relationships among output-class manifolds are precisely reconstructed. These results provide an interpretable, empirically verifiable theoretical foundation and analytical toolkit for understanding representation learning and generalization in DNNs.